Multi-Mechanism Learning for Dynamic Tasks

Physical learning systems are typically studied in the context of static recognition tasks, where parameters are tuned through intrinsic dynamics. Yet such systems are inherently dynamical and can, in principle, learn time-dependent behaviors. Here, we extend the Multi-Mechanism Learning (MML) framework to train spring networks to reproduce desired periodic trajectories. The network consists of input, target, and hidden springs, with the hidden springs modifying their spring constants. Learning is achieved using periodic inputs and adjoint feedback dynamics, which produce a strictly local update rule: each hidden spring constant evolves according to a time-integrated overlap between forward and feedback strains, eliminating the need for global information. To increase task capacity, we further introduce hysteretic springs characterized by double-well potentials. These bistable elements allow the network to encode and retrieve multiple dynamical behaviors. Depending on initial conditions and the shape of the double-well potential, the system can switch between distinct learned trajectories. Together, this demonstrates that physical networks with nonlinear and hysteretic components can learn, store, and flexibly execute multiple dynamical tasks through purely local adaptation.